On Row Sequences of Hermite–Padé Approximation and Its Generalizations

Abstract

Hermite−Padé approximation has been a mainstay of approximation theory since the concept was introduced by Charles Hermite in his proof of the transcendence of e in 1873. This subject occupies a large place in the literature and it has applications in different subjects. Most of the studies of Hermite−Padé approximation have mainly concentrated on diagonal sequences. Recently, there were some significant contributions in the direction of row sequences of Type II Hermite−Padé approximation. Moreover, various generalizations of Type II Hermite−Padé approximation were introduced and studied on row sequences. The purpose of this paper is to reflect the current state of the study of Type II Hermite−Padé approximation and its generalizations on row sequences. In particular, we focus on the relationship between the convergence of zeros of the common denominators of such approximants and singularities of the vector of approximated functions. Some conjectures concerning these studies are posed.

Keywords

• montessus de ballore theorem
• orthogonal polynomials
• faber polynomials
• simultaneous padé approximation
• hermite–padé approximation
• multipoint padé approximation
• inverse type results